Radar system having a beamless emission signature

ABSTRACT

A radar system having a beamless emission signature is described. In one implementation, the radar system includes a transmission system and a receiver system. The transmission system is configured to transmit a pattern comprising a plurality of radar signals having different frequencies simultaneously. The receiver system is configured to receive a reflection of the pattern and combine the plurality of radar signals into a composite waveform forming an image of a target.

TECHNICAL FIELD

The present invention relates generally to phased array radar systems.

BACKGROUND

In a conventional phased array radar system, all elements of an antennasurface radiate and receive identical radar signals at the same time.There may be an adjusted amplitude and/or phase shift between the radarsignals to account for direction of the antennas, but otherwise eachactive element of the antenna surface transmits and receives the sameradar signal (also referred to interchangeably herein as a radar beam).Accordingly, when the radar system is in a transmit mode, a compositesignature waveform of the radar beams is susceptible to observation froma hostile observer, making it possible to jam, evade, and/or somehowinterfere with the radar system.

SUMMARY

A radar system having a beamless emission signature is described. In oneimplementation, the radar system includes a transmission system and areceiver system. The transmission system is configured to transmit apattern including a plurality of radar signals having differentfrequencies, simultaneously. The receiver system is configured toreceive a reflection of the pattern and combine the plurality of radarsignals into a composite waveform to form an image of a target.

The following description, therefore, introduces the broad concept ofradiating a radar beam (i.e., a pattern) that has no apparent main radarbeam by simultaneously transmitting a plurality of radar signals havingdifferent frequencies, and forming a virtual radar beam upon receipt ofthe radar signals by processing the radar signals within the confines ofthe radar system. Accordingly, at no time is a conventional radar beamemitted, because the plurality of radar signals are emitted at amultitude of different frequencies that appear to a hostile observer toresemble fluctuating and scintillating noise, not only in time but inspace coordinates, as well. This makes it extremely difficult for ahostile observer to jam and/or intercept the radar beam generated by theradar system described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description is described with reference to the accompanyingfigures. In the figures, the left-most digit(s) of a reference numberidentifies the figure in which the reference number first appears.

FIG. 1 is a pictorial diagram illustrating select elements used in asystem environment in which an exemplary radar system of the presentinvention may be deployed.

FIG. 2 illustrates waveforms of virtual radar beams (i.e., syntheticradar beams) of the present invention formed by combining the radarsignals of patterns.

FIG. 3 is a block diagram illustrating various components of anexemplary implementation for a radar system that can be utilized toimplement the inventive techniques described herein.

FIG. 4 shows an antenna surface of a radar system of the presentinvention.

FIG. 5 shows a three-dimensional graph showing the nature of a patternemitted as a waveform from a radar system of the present invention.

FIG. 6 shows an exemplary antenna and its reflector geometry as used inthe present invention.

FIG. 7 is a flow chart illustrating an exemplary method of the presentinvention for transmitting and receiving radar signals from a radarsystem.

DETAILED DESCRIPTION

System Overview

FIG. 1 is a pictorial diagram illustrating select elements used in asystem environment 100 in which an exemplary radar system 102 may bedeployed. Environment 100 represents a terrestrial, aerial, and/orspace-based environment in which radar system 102 may observe objects onthe ground and/or in the atmosphere for military and/or civilianapplications.

In one implementation, it is expected that radar system 102 will be usedin a very large aperture radar system. For example, radar system 102 maybe implemented on a platform, approximately the size of a United Statesfootball field, flown in a low earth space-based orbit. Such a massivesystem may have more than 4,500 antennas (also referred to herein asradiators) used for transmitting and receiving radar signals (describedbeginning with reference to FIG. 3). Of course, this is only one exampleof the type of platform on which radar system 102 may be deployed, andit is possible for the size, type, and number of antennas to vary.

Continuing to refer to FIG. 1, radar system 102 generally includes atransmission system 104 and a receiver system 106 that rely on an arrayof antennas (not shown in FIG. 1) to transmit and receive radar signals.For instance, transmission system 104 is configured to transmit patterns108(1), 108(2), 108(3), . . . , 108(N). A pattern, referred to generallyas reference number 108, comprises a plurality of radar signals F₁, F₂,F₃, F₄, F₅, F₆, F₇, . . . , F_(N) transmitted simultaneously in anygiven period of time (e.g., Time (A), Time (A-1), Time (A-2), Time(A-3), and so forth). Each radar signal, referred to generally as F_(x),represents an electromagnetic waveform that generally has a uniquefrequency component when compared to other radar signals in the samepattern. In other words, each radar signal F_(x) of pattern 108generally has its own unique frequency. It is also possible, however,that some of the radar signals F₁, F₂, F₃, F₄, F₅, F₆, F₇, . . . , F_(N)in any particular pattern may have identical frequency waveforms. Forexample, in pattern 108(1), radar signal F₁ and radar signal F₃ may haveidentical waveforms, while the rest of the radar signals, F₂, F₄, F₅,F₆, F₇, . . . , F_(N), of pattern 108(1) may have different (i.e.,unique) waveforms.

Transmission system 104 is capable of transmitting the same pattern eachperiod of time or may vary the pattern by altering the radiatedwaveforms associated with each radar signal over time. By permuting thefrequencies associated with radar signals from pulse-to-pulse (e.g.,Time (A)-to-Time (A- 1), etc.), each successive pattern 108(1), 108(2),. . . 108(N) should appear to a hostile observer to resemble fluctuatingand scintillating noise. Accordingly, at no time is there an emission ofa conventional radar beam from radar system 102. This makes it extremelydifficult for the hostile observer to intercept and jam radar system102. In this fashion, radar system 102 is able to transmit radar signalsstealthily and without an identifiable main radar beam that is referredto as a “beamless emission signature.”

Receiver system 106 is configured to receive a reflection of each of thepatterns 108 reflected from a target 110. With each received pattern108, receiver system 106 is further configured to combine the radarsignals F₁, F₂, F₃, F₄, F₅, F₆, F₇, . . . , F_(N) of each pattern 108 toform a composite aggregate waveform representing an image of target 110.

For example, FIG. 2 illustrates waveforms 202(1) and 202(N) that arevirtual radar beams (i.e., synthetic radar beams) formed by combiningeach of the radar signals of patterns 108(1) and 108(N), respectively.In other words, receiver system 106 receives reflected patterns 108(1)and 108(N) from a target 110 (FIG. 1) and converts each pattern into awaveform 202(1) and 202(N), respectively, representing a virtual radarbeam. Accordingly, radar system 102 (FIG. 1) is able to form an image ofa target 110 (FIG. 1) by processing each of the waveforms 202. Byforming a virtual radar beam within the confines of radar system 102(FIG. 1), it is possible to avoid transmitting and receiving aconventional main radar beam with physical properties that areobservable.

FIG. 3 is a block diagram illustrating various components of anexemplary implementation for radar system 102 that can be utilized toimplement the inventive techniques described herein. Radar system 102includes, a common reference oscillator 301, transmit/receive (T/R)elements 302(1), 302(2), . . . , 302(N), with their signal processing304(1), 304(2), . . . , 304(N), and control processor 316.

Common reference oscillator 301 is a fixed oscillator circuit or avariable oscillator, such as a voltage controlled oscillator (VCO).Common reference oscillator 301 is used to create a particular referencesignal 305 that serves as a common unmodulated pilot signal for T/Relements, referred to generally as reference number 302. In theexemplary implementation, common reference oscillator 301 relies on oneor more crystals (not shown) to generate the reference signal 305.

Common reference signal 305 can be distributed from common referenceoscillator 301 via a coaxial cable or a microstrip line. Alternatively,common reference oscillator 301 may be positioned in the middle of anarray of antenna elements and distribute a low power (e.g., 1 mW)signal. For example, FIG. 4 shows an antenna surface 402 representing asubstrate or base supporting a plurality of antennas of a radar system102. In particular, antenna surface 402 includes radiators 404, and thecommon reference oscillator 301 located in the middle. Each of theradiators 404 are individually phased locked to the common referenceoscillator 301 that generates an unmodulated reference pilot and thusthe radiators 404 (via their constituent elements, such as T/R elements302 and signal processors 304, etc.), are kept phase coherent with eachother.

Referring to FIG. 3, T/R elements 302 are separate radar units operatingcollectively as a phased array radar system. In one exemplaryimplementation, each T/R element 302 includes a transmission unit 322and a receiver unit 324. In particular, each of the transmission units322 generally includes a phase locked loop (PLL) 306, a phase andamplitude modulator 308, and a transmit amplifier 310. Each of thereceiver units 324 generally includes a processor 304, a receiveamplifier 312, and an analogue-to-digital converter (ADC) 314. Bothtransmission units 322 and receiver units 324 also generally utilize anantenna duplexer 318, and a radiator 320.

PLLs 306 generally include a local radio frequency (RF) oscillator notshown, but which is readily understood by those skilled in the art. PLLs306 assist in creating a particular carrier frequency for signals to betransmitted by each radiator 320. In the exemplary implementation, eachPLL 306 is phase-locked to reference signal 305. Alternatively, a directdigital synthesizer (not shown) with a combination of anupconverter/fixed frequency oscillator (not shown) can also be employed.

Phase and amplitude modulators 308 vary the phase and amplitude ofsignals generated by PLLs 306. Alternatively, phase modulators can alsobe incorporated within the phase locked loops 306, or completelyarbitrarily modulated waveforms can be generated by direct digitalsynthesis. Transmit amplifiers 310 amplify signals produced by PLLs 306and phase shifters 308 for transmission by each antenna 320. Antennaduplexers 318 enable signals to be sent and received by a radiator 320.For a pulse radar system, antenna duplexers 318 can be implemented as atransmit/receive switch. Alternatively, antenna duplexers 318 can beimplemented as circulator or other type of antenna switch configuration.

Radiators 320 are used as a conduit for receiving and/or transmittingsignals. Some radar systems may use more than one radiator per T/Relement 302 for transmitting or receiving signals. For purposes, of thisdiscussion, it should be appreciated that radiator 320 is identical toradiators 404 shown in FIG. 4, albeit shown in more detail withreference to T/R elements 302 in FIG. 3. It is understood that eachphysical radiator 320/404 (FIGS. 3/4), is driven by electronic elementssuch as T/R elements 302, etc. It should also be recognized thatradiators 320 come in a variety of forms, and for purposes of thisdiscussion any of these variety of forms may be included.

Referring to FIG. 3, receive amplifiers 312 amplify signals received bya radiator 320 and transmit the amplified signals to ADCs 314 forconversion to a digital domain. It should be recognized that amplifiersand ADCs come in a variety of configurations, and for purposes of thisdiscussion any of these variety of configurations may be included.

Processors 304 execute various instructions to control the operation ofeach T/R element 302 and to communicate with a control processor 316. Inone implementation, processors 304 and control processor 316 areimplemented as digital signal processors. In other implementations,processors may be implemented as self-executing programmable logicarrays in the form of a FPGA (Field Programmable Gate Array),microprocessors, one or more ASICs (Application Specific IntegratedCircuit), or other hardware-based technology including hybrid circuitand programmable logic technologies. It is also to be appreciated thatthe components and processes described herein can be implemented insoftware, firmware, hardware, or combinations thereof. By way ofexample, a DSP, programmable logic device (PLD) or ASIC could beconfigured or designed to implement various components and/or processesdiscussed herein.

Processors 304 generally rely on control processor 316 for housekeepingoperations, such as initializing radar system 102, handling errorconditions, and other duties that generally fall outside the scope ofeach respective processor 304.

Those skilled in the art will recognize that there are many differenttypes of transmission units 322 and receiver units 324, and that for thepurposes of this discussion, most receivers and transmitters may includeany of these different types. It is to be appreciated that additionalcomponents can be included in each T/R element 302 and some componentsillustrated in each T/R element 302 above need not be included. Forexample, additional processors 304 may be included in a T/R element 302,or phase and amplitude modulators 308 may not be included.

Having introduced various components of radar system 102, it is nowpossible to describe its relevant operation in more detail withreference to FIG. 3.

Each T/R element 302 is configured to transmit a radar signal with afrequency F_(x) that is generally unique to the particular antennaelement. Each T/R element 302 is configured to receive a conglomerationof the radar signals F₁, F₂, F₃, F₄, F₅, F₆, F₇, . . . , F_(N) (FIG. 1)reflected from a target 110 (FIG. 1), including the particular radarsignal with a frequency F_(x) unique to the particular antenna element.

For example, suppose at a time period A, T/R element 302(1) transmits aradar signal (via one or more radiators 320) associated with aparticular pattern having a frequency of F₃, T/R element 302(2)transmits a radar signal (associated with the same pattern) having afrequency of F₅, and T/R element 302(N) transmits a radar signal with afrequency of F₁. At a period of time later (such as double the transmittime) T/R elements 302(1), 302(2), . . . , 302(N) will receive (viaradiators 320) a conglomeration of the radar signals F₁, F₂, F₃, F₄, F₅,F₆, F₇, . . . , F_(N) reflected form the target 110 (FIG. 1) in the formof a pattern 108 (FIG. 1), including the frequency that is unique to theparticular antenna element: i.e., in this example, radar signal F₃ isunique to T/R element 302(1); radar signal F₅ is unique to T/R element302(2); and radar signal F₁ is unique to T/R element 302(N).

Each receive processor 304, is configured to correlate the compositereflected radar signal that was generated locally by the particular T/Relement 302. For instance, processor 304(1) is configured to determinewhich of the radar signals F₁, F₂, F₃, F₄, F₅, F₆, F₇, . . . , F_(N)reflected from the target was locally generated by T/R element 302(1).Likewise, processor 304(2) is configured to recognize which of the radarsignals F₁, F₂, F₃, F₄, F₅, F₆, F₇, . . . , F_(N) reflected form thetarget was locally generated by T/R element 302(2), and so forth.

Based on each of the locally recognized radar signals by processors 304,control processor 316 is able to correlate the radar signals to the T/Relements 302 that transmitted the radar signals, and combine the radarsignals to form an image of the target 110 (FIG. 1). Accordingly,processor 304 and control processor 316 act as processing systemenabling a reflected composite signal to be demodulated, or correlated,at each T/R element 302, and combined. This coherent correlation overtime enables processing system 304/316 to form a synthetic and spatiallyselective antenna beam.

Having introduced radar system 102, it is now possible to describe itsfeatures, analytical details, and operation in more detail.

Theoretical and Analytical Details

In a conventional “static” phased array, all elements of the antennasurface radiate the same time waveform except for its amplitude andphase. The appropriate setting of these amplitudes and phases will forman antenna radiation pattern whose shape is independent of the timeevolution of the waveform as long as the latter is “narrowband”, thatis, its bandwidth is much less than the carrier frequency. At any giventime, the radiation pattern is the Fourier transform of the aperturedistribution. In synthetic aperture radar, where a single radiator ismoved to cover an effective antenna surface while the waveforms radiatedat different points and instances are identical from pulse to pulse, thesituation is similar. Because of the complete linearity and timeinvariance of electromagnetic wave radiation and propagation aparticular waveform can be processed by sampling it at various times,frequencies and locations. As long as all relevant samples are “covered”a coherent receiver should be able to reconstruct the image.

In contrast to a conventional synthetic aperture radar, radar system 102decomposes the desired time waveform into its frequency components, thentransmits each frequency component from one or several T/R elements 302(FIG. 3).

Upon reflection, the received waveform will contain the composite of allthe transmitted frequencies. While there is a single reflected waveform,the individual radiators 320 will receive one a signal whose phasevaries with the element's location. Since at any given time radiators320 generally transmit at different frequencies, in the conventionalsense there is no “real” beam being formed during transmission of theradar pulse. In other words, the resulting interference pattern is notstatic but fluctuates at the envelope rate of the composite waveform,but upon coherently processing the reflected waveform, a syntheticantenna radiation pattern is being formed when the phases of thereceived signal are properly aligned.

FIG. 5 shows the three-dimensional nature of a pattern 108 emitted as awaveform 502 from radar system 102. Waveform 502 includes a frequencydimension, a time dimension, and a radiating element index dimension.Each sphere 504 represents an elementary waveform, i.e., a pulse at aparticular frequency, as transmitted by a radiator 320 (also referred toas a radiating element).

FIG. 6 shows an exemplary antenna surface 402 and element 320 and apoint reflector. For purposes of discussion, assume that radiator 320(FIG. 6) is referred to as radiating element k. Also assume that k islocated on the antenna surface 402 at a point {overscore (α)}_(k) and apulse of length T_(p) is transmitted with phase σ_(k), amplitude A_(k)and frequency u_(k). The particular phase shifter settings σ_(k) controlthe antenna scanning, i.e., focusing or collimating. The antenna surface402 of an antenna does not have to be flat, thus the concept applies toso-called conformal antennas, as well.

In complex envelope notation, the composite pulse transmitted by theantenna in the far field is proportional to:

$\begin{matrix}{{E\left( {\overset{\rightarrow}{R},t} \right)} \propto {\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{{rect}\left( \frac{t - \tau_{k}^{\prime}}{T_{p}} \right)}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau_{k}^{\prime}})}}}}}} & (0.1)\end{matrix}$

where the summation runs over the antenna elements indexed by k;

$\tau_{k}^{\prime} = \frac{{\overset{\rightarrow}{R} - {\overset{\rightarrow}{a}}_{k}}}{c}$is the time it takes for the EM wave to reach the reflector located atpoint {overscore (R)} when it is launched from the radiator at{overscore (α)}_(k) and we define the rectangular pulse as rect

$(t) = {{1\mspace{14mu}{for}\mspace{14mu}{t}} < \frac{1}{2}}$and 0 otherwise.

Denoted by θ, an arbitrary off-boresight angle and by ψ_(k) the angle ofthe k^(th) radiator. (If the antenna surface is flat and the origin ofcoordinate system is on it then the angles, then ψ_(k)=0 or π.) Far fromthe antenna, in the far field we have the approximation

$\begin{matrix}{\tau_{k}^{\prime} = {{{\frac{1}{c}\sqrt{R^{2} + a_{k}^{2} - {2{\overset{\rightarrow}{R} \cdot {\overset{\rightarrow}{a}}_{k}}}}} \approx {\frac{R}{c}\left( {1 - {\frac{1}{R}{{\overset{\rightarrow}{R}}^{0} \cdot {\overset{\rightarrow}{a}}_{k}}}} \right)}} = {\tau^{\prime} - {\frac{a_{k}}{c}{\sin\left( {\theta - \psi_{k}} \right)}}}}} & (0.2)\end{matrix}$

$\tau^{\prime} = \frac{R}{c}$being the effective one-way delay. The aggregate field at a given pointof space {overscore (R)}(τ′,θ) represented by the delay τ′ andoff-boresight direction θ is

${E\left( {\overset{\rightarrow}{R},t} \right)} \propto {\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\sigma_{k}}{{rect}\left( \frac{t - \tau_{k}^{\;^{\prime}}}{T_{p}} \right)}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau_{k}^{\prime}})}}}}} \approx {\sum\limits_{k}{A_{k\;}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{{rect}\left( \frac{t - \tau_{k}^{\prime}}{T_{p}} \right)}{\exp\left\lbrack {{\mathbb{i}2\pi}\;{u_{k}\left( {t - \tau^{\prime} + {\frac{a_{k}}{c}{\sin\left( {\theta - \psi_{k}} \right)}}} \right)}} \right\rbrack}}}$

and thus

$\begin{matrix}{{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau_{k}^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\sigma_{k}}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau^{\prime}})}}}{\exp\left\lbrack {{\mathbb{i}2\pi}\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta - \psi_{k}} \right)}} \right\rbrack}}}}} & (0.3)\end{matrix}$

The last approximation step follows because the rectangular pulse rect

$\left( \frac{t}{T_{p}} \right)$is much longer than any variation in τ′_(k) over the antenna surface:

${{rect}\left( \frac{t - \tau_{k}^{\prime}}{T_{p}} \right)} \approx {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}.}$

Assume that a point reflector is located at {overscore (R)}(τ′,θ_(r)):

$\begin{matrix}{{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta_{r}} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\alpha_{k}}{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\;{u_{k}{({t - \tau^{\prime}})}}}{\exp\left\lbrack {{\mathbb{i}}\; 2\;\pi\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta_{r} - \psi_{k}} \right)}} \right\rbrack}}}}} & (0.4)\end{matrix}$

If the phase shifters are set so that

$\begin{matrix}{\sigma_{k} \equiv {{- 2}\pi\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta_{r} - \psi_{k}} \right)}{mod}\mspace{14mu} 2\pi}} & (0.5)\end{matrix}$

Then

${{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\exp\left\lbrack {{\mathbb{i}2\pi}\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta_{r} - \psi_{k}} \right)}} \right\rbrack}} = 1$independently of the target range, and from (0.4) for the wave amplitudeis obtained in the “focused” or collimated direction

$\begin{matrix}{{{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta_{r}} \right)},t} \right)} \propto {\sum\limits_{k}{A_{k}{{rect}\left( \frac{t - \tau_{k}^{\prime}}{T_{p}} \right)}{\mathbb{e}}^{{\mathbb{i}2\pi u}_{k}{({t - \tau^{\prime}})}}}} \approx {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau^{\prime}})}}}}}}},} & (0.6)\end{matrix}$

that is the time delayed version of rect

$\left( \frac{t}{T_{p}} \right){\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}2\pi}\; u_{k}t}}}$the latter being the desired composite radar signal.

The appropriate setting σ_(k) for each phase shifter depends, of course,on the desired off-boresight angle θ_(r). In the simplest case, when theantenna surface is flat, sin ψ_(k)=0, and the radiators are uniformlyspaced, α_(k)=kd, one obtains

$\sigma_{k} \equiv {{- 2}\pi\frac{kd}{\lambda_{k}}}$sin θ_(r), a standard result for beam steering when all the frequenciesu_(k)=c/λ_(k) are also set to be the same.Radiation Pattern

For any given setting of the phase shifters and amplitudes the fieldE({overscore (R)}(τ′,θ),t) in an arbitrary direction is given by (0.3).

When the phase shifters are set to collimate the beam in the directionrepresented by the off-boresight angle θ_(r),

where

$\begin{matrix}{{\sigma_{k} \equiv {{- 2}\;\pi\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta_{r} - \psi_{k}} \right)}\mspace{14mu}{mod}{\mspace{11mu}\;}2\;\pi}},} & (0.5)\end{matrix}$sin (θ_(r)−ψ_(k)) mod 2π(0.5), and the field E({overscore (R)}(τ′,θ),t)in any other arbitrary direction θ, is determined as follows:

${{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau^{\prime}})}}}{\exp\left\lbrack {{\mathbb{i}2\pi}\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta - \psi_{k}} \right)}} \right\rbrack}}}}} = {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau^{\prime}})}}}{\exp\left\lbrack {{\mathbb{i}}\; 2\;\pi\; u_{k}\frac{a_{k}}{c}\left( {{\sin\left( {\theta - \psi_{k}} \right)} - {\sin\left( {\theta_{r} - \psi_{k}} \right)}} \right)} \right\rbrack}}}}$

or

$\begin{matrix}{{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\; 2\pi\;{u_{k}{({t - \tau^{\prime}})}}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}\frac{a_{k}}{c}2{\sin{({\frac{1}{2}{({\theta - \theta_{r}})}})}}{\cos{({{\frac{1}{2}{({\theta + \theta_{r}})}} + \psi_{k}})}}}}}}} & (1.1)\end{matrix}$

When θ≈θ_(r), the approximation is

${E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\;{u_{k}{({t - \tau^{\prime}})}}}\left( {1 + {{\mathbb{i}2\pi}\; u_{k}\frac{a_{k}}{c}{\cos\left( {\theta_{r} + \psi_{k}} \right)}\left( {\theta - \theta_{r}} \right)}} \right)}}}}$${{E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} - {E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)}} \propto {\left( {\theta - \theta_{r}} \right){rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right){\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau^{\prime}})}}}{\mathbb{i}2\pi}\; u_{k}\frac{a_{k}}{c}{\cos\left( {\theta_{r} + \psi_{k}} \right)}}}}$That is, the field intensity is nearly linear function of the angle forsmall deviation from the nominal boresight.

If this were conventional radar, then the radiation pattern E({overscore(R)}(τ′,θ),t) would be proportional to the transmitted pulse rect

${\left( \frac{t - \tau^{\prime}}{T_{p}} \right){\mathbb{e}}^{{\mathbb{i}\pi}\;{u{({t - \tau^{\prime}})}}}},$where the shape of the proportionality factor F(θ) would depend only onthe locations of the radiating elements, as defined by

${F(\theta)} = {\sum\limits_{k}{\mathbb{e}}^{{\mathbb{i}2\pi}\; u\frac{a_{k}}{c}2{\sin{({\frac{1}{2}{({\theta - \theta_{r}})}})}}{\cos{({{\frac{1}{2}{({\theta + \theta_{r}})}} + \psi_{k}})}}}}$with |F(θ)|² being the antenna directivity.

Since in (1.1) such factorization is impossible, there is no transmitantenna pattern in a conventional sense. Instead the pattern will bevarying with time, and fluctuate.

Received Wave from the Collimated Direction

Upon reflection from the point target at {overscore (R)}, the receivedfield at T/R element 302 m will be proportional to, see (0.6):

$\begin{matrix}{{E_{m}\left( {\overset{\rightarrow}{R},t} \right)} \propto {E\left( {\overset{\rightarrow}{R},{t - \tau_{m}^{\prime}}} \right)} \approx {{{rect}\left( \frac{t - \tau^{\prime} - \tau_{m}^{\prime}}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau_{m}^{\prime}})}}}}}}} & (2.1)\end{matrix}$

Notice that this signal contains all the transmitted frequencies. Toreceive it optimally, one must, therefore, correlate it with its locallyregenerated complex conjugate Ē_(m)({overscore (R)},t) and then sum theresulting correlations over the several receivers.

${{\tau_{m}^{\prime} \approx {\tau^{\prime} - {\frac{a_{m}}{c}{\sin\left( {\theta_{r} - \psi_{m}} \right)}}}} = {\tau^{\prime} - {\delta\tau}_{m}}},$is a delay dependent on the radiating element and on the beam direction,but is independent of the target range, 2πu_(m)δτ_(m)≡−α_(m) mod 2π.Denoting the nominal roundtrip propagation delay between the radar andthe target by τ=2τ′, the received field intensity at receiver m isproportional to:

${{E_{m}\left( {\overset{\rightarrow}{R},t} \right)} \propto {E\left( {\overset{\rightarrow}{R},{t - \tau_{m}^{\prime}}} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime} - \tau_{m}^{\prime}}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}}\; 2\pi\;{u_{k}{({t - \tau^{\prime} - \tau_{m}^{\prime}})}}}}}} \approx {{{rect}\left( \frac{t - {2\tau^{\prime}}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau^{\prime} - \tau^{\prime} + {\delta\tau}_{m}})}}}}}}} = {{{rect}\left( \frac{t - \tau}{T_{p}} \right)}{\sum\limits_{k}{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}{\delta\tau}_{m}}{\mathbb{e}}^{{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau})}}} =}{V_{m}\left( {t - \tau} \right)}}}}$That is:E _(m)({overscore (R)},t)∝V _(m)(t−τ)  (2.2)

Besides being dependent both on the transmitted frequencies and on thereceiving module m, the time function

$\begin{matrix}{{V_{m}(t)} = {{{rect}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}{\delta\tau}_{m}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}t}}}}} & (2.3)\end{matrix}$

through the relationship

${{\frac{a_{m}}{c}{\sin\left( {\theta_{r} - \psi_{m}} \right)}} = {\delta\tau}_{m}},$also depends implicitly on the direction the beam is being collimatedbut does not depend on the range delay τ.

Next, the correlation is calculated

${K_{m}(s)} = {{\frac{1}{T_{p}}{\int_{- \infty}^{\infty}{{E_{m}\left( {\overset{\rightarrow}{R},t^{\prime}} \right)}{\overset{\_}{E\ }}_{m}\left( {\overset{\rightarrow}{R},{s + \tau + t^{\prime}}} \right){\mathbb{d}t^{\prime}}}}} = {\frac{1}{T_{p}}{\int_{- \infty}^{\infty}{{E_{m}\left( {\overset{\rightarrow}{R},t^{\prime}} \right)}{{\overset{\_}{V}}_{m}\left( {s + t^{\prime}} \right)}\ {\mathbb{d}t^{\prime}}}}}}$

Here the appearance of s=t−τ in the argument of V_(m) signifies that therange gate of the radar receiver is centered around the epoch τ, i.e.,the “running time” is t=s+τ.

$\begin{matrix}{{K_{m}(t)} = {{\frac{1}{T_{p}}{\int_{- \infty}^{\infty}{{E_{m}\left( {\overset{\rightarrow}{R},t^{\prime}} \right)}{{\overset{\_}{V}}_{m}\left( {t - \tau + t^{\prime}} \right)}\ {\mathbb{d}t^{\prime}}}}} \approx \begin{bmatrix}{{\sum\limits_{k}\;{{A_{k}}^{2}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\; u_{k}t}\frac{1}{T_{p}}{\int_{\frac{T_{p}}{2} + \tau}^{\frac{T_{p}}{2} + \tau}{{{rect}\left( \frac{t + t^{\prime} - \tau}{T_{p}} \right)}\ {\mathbb{d}t^{\prime}}}}}} +} \\{\sum\limits_{\underset{k \neq k^{\prime}}{k}}\;{\sum\limits_{k^{\prime}}{A_{k}{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\;{({u_{k} - u_{k^{\prime}}})}{({\tau - {\delta\tau}_{m}})}}\frac{1}{T_{p}}{\int_{\frac{T_{p}}{2} + \tau}^{\frac{T_{p}}{2} + \tau}{{{rect}\left( \frac{t + t^{\prime} - \tau}{T_{p}} \right)}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}t^{\prime}}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\;{u_{k^{\prime}}{({t + t^{\prime}})}}}{\mathbb{d}t^{\prime}}}}}}}\end{bmatrix}}} & (2.4)\end{matrix}$

This formula contains simple integrals that can be evaluated by anappropriate change of variable.

Therefore,

${K_{m}(t)} \approx {{\sum\limits_{k}\;{{A_{k}}^{2}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\; u_{k}t}{{tria}\left( \frac{t}{T_{p}} \right)}}} + {{cross}{\mspace{11mu}\;}{terms}\mspace{14mu}\left( {k,k^{\prime}} \right)}}$

By referencing the time to the instant the pulse echo arrives at theantenna we have shifted the argument of the correlation function by τ.The epoch t=0 is thus especially important, for it represents the centerof the range gate, that is, the moment of peak detection:

${K_{m}(0)} \approx {{\sum\limits_{k}\;{A_{k}}^{2}} + {{cross}\mspace{14mu}{terms}\mspace{14mu}{\left( {k,k^{\prime}} \right).}}}$cross terms (k,k′). At t=0 the double sum of the cross terms alsosimplifies to:

$\sum\limits_{\underset{k \neq k^{\prime}}{k}}{\sum\limits_{k^{\prime}}{A_{k}{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{{\mathbb{i}2\pi}{({u_{k} - u_{k^{\prime}}})}}{\delta\tau}_{m}}\sin\;{c\left( {\left( {u_{k} - u_{k^{\prime}}} \right)T_{p}} \right)}}}$

Notice that to collimate the receive beam during receive processing, itis not necessary to have to employ a pair of attenuation and phaseshifter settings, A_(m)′ and α_(m)′ explicitly. In fact, this gainsnothing because those beam parameters were implicitly part of theconstruction of the locally regenerated composite signal

${V_{m}(t)} = {{{rect}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}{\delta\tau}_{m}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}t}}}}$with which is correlated the reflected wave.

The final compressed signal is the sum over the M(m=1,2, . . . , M)receivers of their individual “partial” correlations:

${z(t)} = {{\sum\limits_{m}\;{K_{m}(t)}} \approx {\sum\limits_{m}\;\left\{ {{\sum\limits_{k}{{A_{k}}^{2}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\; u_{k}t}{{tria}\left( \frac{t}{T_{p}} \right)}}} + {{cross}\mspace{14mu}{terms}\mspace{14mu}\left( {k,k^{\prime}} \right)}} \right\}}}$

When t=0

${z(0)} = {M{\sum\limits_{k}{{A_{k}}^{2}\underset{k^{\prime}}{+ \sum}{\sum\limits_{\underset{k \neq k^{\prime}}{k}}\left\{ {A_{k}{\overset{\_}{A}}_{k^{\prime}}\sin\;{c\left( {\left( {u_{k} - u_{k^{\prime}}} \right)T_{p}} \right)}{\sum\limits_{m}{\mathbb{e}}^{{\mathbb{i}2}\;{\pi{({u_{k} - u_{k^{\prime}}})}}{\delta\tau}_{m}}}} \right\}}}}}$

From

${\frac{a_{m}}{c}{\sin\left( {\theta_{r} - \psi_{m}} \right)}} = {\delta\tau}_{m}$the inner sum is

${\sum\limits_{m}{\mathbb{e}}^{{\mathbb{i}2}\;{\pi{({u_{k} - u_{k^{\prime}}})}}{\delta\tau}_{m}}} = {\sum\limits_{m}{{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{({u_{k} - u_{k^{\prime}}})}\frac{a_{m}}{c}{\sin{({\theta_{r} - \psi_{m}})}}}.}}$

For a football field size radar under consideration α_(m)≦300 m and|u_(k)–u_(k′)|≦100 MHz, therefore, the argument of the exponential isbounded by

${{{{{u_{k} - u_{k^{\prime}}}}\frac{a_{m}}{c}} \leq {100\mspace{14mu}{MHz}\frac{300\mspace{14mu} m}{300\mspace{14mu} m\text{/}\mu\;\sec}}} = 100},$representing a large number of cycles. Therefore, it is expected thatthis sum will fluctuate near zero for all off-boresight target angles.This natural averaging of the unwanted terms is similar to the one inspread spectrum systems, the difference here being simultaneouslyaveraged both in time and in space domains.Receive Antenna Pattern

Next, is a discussion directed to calculating the full antenna patternin receive mode. The field E({overscore (R)}(τ′,θ),t) at an arbitrarynominal delay τ′ and direction θ is reflected back to the m^(th) antennaelement where it is coherently correlated with the local reference

${V_{m}(t)} = {{{rect}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\; 2\pi\; u_{k}{\delta\tau}_{0m}}{{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\; u_{k}t}.}}}}$The phase shifters are set to collimate in direction θ_(r), that is

${\delta\tau}_{0m} = {\frac{a_{m}}{c}{{\sin\left( {\theta_{r} - \psi_{m}} \right)}.}}$

In an arbitrary direction θ≠θ_(r), though,

${\tau_{m}^{\prime} \approx {\tau^{\prime} - {\frac{a_{m}}{c}{\sin\left( {\theta - \psi_{m}} \right)}}}} = {\tau^{\prime} - {\delta\tau}_{m}}$with

${{\delta\tau}_{m} = {\frac{a_{m}}{c}{\sin\left( {\theta - \psi_{m}} \right)}}},$and then (1.1)

${E\left( {{\overset{\rightarrow}{R}\left( {\tau^{\prime},\theta} \right)},t} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t - \tau^{\prime}})}}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}\frac{uk}{c}{\sin{({\theta - \psi_{k}})}}}}}}$

from which the received wave at the m^(th) antenna element is

${E_{m}\left( {\overset{\rightarrow}{R},t} \right)} \propto {E\left( {\overset{\rightarrow}{R},{t - \tau_{m}^{\prime}}} \right)} \propto {{{rect}\left( \frac{t - \tau^{\prime} + \tau_{m}^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau^{\prime} - \tau_{m}^{\prime}})}}}{\mathbb{e}}^{{\mathbb{i}2\pi}\frac{a_{k}}{\lambda_{k}}{\sin{({\theta - \psi_{k}})}}}}}} \approx {{{rect}\left( \frac{i - {2\tau^{\prime}} + {\delta\tau}_{m}^{\prime}}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - {2\tau^{\prime}} + {\delta\tau}_{m}^{\prime}})}}}{\mathbb{e}}^{{\mathbb{i}2\pi}\frac{a_{k}}{\lambda_{k}}{\sin{({\theta - \psi_{k}})}}}}}} \approx {{{rect}\left( \frac{t - \tau}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}\sigma}_{k}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau + {\delta\tau}_{m}})}}}{\mathbb{e}}^{{\mathbb{i}2\pi}\frac{a_{k}}{\lambda_{k}}{\sin{({\theta - \psi_{k}})}}}}}}$

but since θ≠θ_(r), at no time is it expected that E_(m)({overscore(R)},t) be proportional to V_(m)(t−τ).

To simplify the formulas let:

${\mu_{k}(\theta)} = {{\sigma_{k} + {2\;\pi\; u_{k}\frac{a_{k}}{c}{\sin\left( {\theta - \psi_{k}} \right)}}} = {{2\;\pi\; u_{k}\frac{a_{k}}{c}\left( {{\sin\left( {\theta - \psi_{k}} \right)} - {\sin\left( {\theta_{r} - \psi_{k}} \right)}} \right)} = {2\pi\;{u_{k}\left( {{\delta\;\tau_{k}} - {\delta\tau}_{0k}} \right)}}}}$

Then the cross-correlation of

${E_{m}\left( {\overset{\rightarrow}{R},t} \right)} \propto {{{rect}\left( \frac{t - \tau}{T_{p}} \right)}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\mu_{k\;}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k}{({t - \tau^{\prime} + {\delta\tau}_{m}^{\prime}})}}}}}}$with V_(m)(t−τ) is

${K_{m}\left( {\overset{\rightarrow}{R},t} \right)} = {{{\frac{1}{T_{p}}{\int_{- \infty}^{\infty}{{E_{m}\left( {\overset{\rightarrow}{R\ },t^{\prime}} \right)}{{\overset{\_}{V}}_{m}\left( {t - \tau^{\prime} + t^{\prime}} \right)}{\mathbb{d}t^{\prime}}}}} \propto {\frac{1}{T_{p}}{\int_{- \infty}^{\infty}{{rect}\frac{\left( {t^{\prime} - \tau} \right)}{T_{p}}{\sum\limits_{k}\;{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\mu_{k}}{\mathbb{e}}^{{\mathbb{i}2\pi}\;{u_{k}{({t^{\prime} - \tau + {\delta\tau}_{m}^{\prime}})}}}{{rect}\left( \frac{t + t^{\prime} - \tau}{T_{p}} \right)}{\sum\limits_{k^{\prime}}\;{{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\; u_{k^{\prime}}{\delta\tau}_{0m}}{\mathbb{e}}^{{- {\mathbb{i}2}}\;\pi\;{u_{k^{\prime}}{({t + t^{\prime} - \tau})}}}{\mathbb{d}t^{\prime}}}}}}}}}} = {\sum\limits_{k}{\sum\limits_{k^{\prime}}{A_{k}{\mathbb{e}}^{{\mathbb{i}}\;\mu_{k}}{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{- {\mathbb{i}}}\; 2{\pi{({u_{k} - u_{k^{\prime}}})}}\tau}{\mathbb{e}}^{{\mathbb{i}}\; 2\;{\pi{({u_{k}{\delta\tau}_{0m}})}}}\frac{1}{T_{p}}{\int_{\frac{T_{p}}{2} + \tau}^{\frac{T_{p}}{2} + \tau}{{{rect}\left( \frac{t + t^{\prime} - \tau}{T_{p}} \right)}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\; u_{k}t^{\prime}}{\mathbb{e}}^{{\mathbb{i}2}\;\pi\;{u_{k^{\prime}}{({t + i^{\prime}})}}}\ {\mathbb{d}t^{\prime}}}}}}}}$

When all the correlations from the individual receivers are combinedcoherently, the compressed signal is the sum of the correlations overall the antenna elements:

${{U\left( {\overset{\rightarrow}{R},t} \right)} \propto {{{tria}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{m}\;{\sum\limits_{k}\;{\sum\limits_{k^{\prime}}\;{A_{k}{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{\mathbb{i}}\; 2\;{\pi{({{u_{k}{({{\delta\;\tau_{k}} - {\delta\tau}_{0k}})}} - {({u_{k}{\delta\tau}_{m - {u_{k^{\prime}}{\delta\tau}_{0m}}}})}})}}}{\mathbb{e}}^{{- {\mathbb{i}}}\;{\pi{({u_{k} + u_{k^{\prime}}})}}t}{\mathbb{e}}^{{- {\mathbb{i}}}\;{\pi{({u_{k} + u_{k^{\prime}}})}}i}{{sinc}\left( {\left( {u_{k -}u_{k^{\prime}}} \right)\left( {T_{p} - {t}} \right)} \right)}}}}}}} = {{{tria}\;\left( \frac{t}{T_{p}} \right){\sum\limits_{k}\;\left\{ {{A_{k}}^{2}{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\;{u_{k}{({{\delta\;\tau_{k}} - {\delta\;\tau_{0k}}})}}}{\mathbb{e}}^{{- {\mathbb{i}}}\; 2\;\pi\; u_{k}t}{\sum\limits_{m}\;{\mathbb{e}}^{{- {\mathbb{i}}}\; 2\;\pi\;{u_{k}{({{\delta\tau}_{m} - {\delta\tau}_{0m}})}}}}} \right\}}} + {{{tria}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{\underset{k \neq k^{\prime}}{k,k^{\prime}}}\;\left\{ {A_{k}{\overset{\_}{A}}_{k^{\prime}}{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\;{u_{k}{({{\delta\;\tau_{k}} - {\delta\;\tau_{0k}}})}}}{\mathbb{e}}^{{\mathbb{i}}\; 2\;{\pi{({u_{k} - u_{k^{\prime}}})}}\tau}\;{\mathbb{e}}^{{- {\mathbb{i}}}\mspace{11mu}\pi\;{u_{k}{({u_{k} - u_{k}})}}i}\sin\;{c\left\lbrack {\left( {u_{k} - u_{k^{\prime}}} \right)\left( {T_{p} - {t}} \right)} \right\rbrack}{\sum\limits_{m}\;{\mathbb{e}}^{{- {\mathbb{i}}}\; 2\;{\pi(\;{{u_{k}{\delta\tau}_{m}} - {u_{k}{\delta\tau}_{0m}}})}}}} \right\}}}}$

If the terms

$\sum\limits_{m}\;{\mathbb{e}}^{{- {\mathbb{i}}}\; 2\;{\pi(\;{{u_{k}{\delta\tau}_{m}} - {u_{k^{\prime}}{\delta\tau}_{0m}}})}}$and especially

$\sum\limits_{m}\;{\mathbb{e}}^{{- {\mathbb{i}}}\; 2\;\pi\;{u_{k}{({{\delta\tau}_{m} - {u_{k}{\delta\tau}_{0m}}})}}}$are small, then the cross-correlation U({overscore (R)},t) will also besmall. This can happen if θ differs from θ_(r) sufficiently enough sothat |u_(k)δτ_(m)−u_(k′)δτ_(0m)|≧1, in which case the exponentials inthe above sum will run over at least a full cycle thereby averaging thecross-correlation U({overscore (R)},t) to zero.

For the special case of uniformly spaced antenna elements on a flatsurface, sin ψ_(k)=0, α_(k)=kd and

${\sigma_{k} \equiv {{- 2}\;\pi\frac{kd}{\lambda_{k}}\sin\;\theta_{r}}},$we can prove more. For then:

${{\delta\tau}_{0m} = {{\frac{a_{m}}{c}{\sin\left( {\theta_{r} - \psi_{m}} \right)}} = {m\frac{d}{c}{\sin\left( \theta_{r} \right)}}}},{{\delta\tau}_{m} = {{\frac{a_{m}}{c}{\sin\left( {\theta - \psi_{m}} \right)}} = {m\frac{d}{c}{\sin(\theta)}}}}$and, thus, the antenna pattern factor can be calculated explicitly

$\begin{matrix}{{{F_{{kk}^{\prime}}(\theta)} = {q_{{kk}^{\prime}}^{- M}\frac{\sin\left\lbrack {M\frac{\pi\; d}{c}\left( {{u_{k}{\sin(\theta)}} - {u_{k^{\prime}}{\sin\left( \theta_{r} \right)}}} \right)} \right\rbrack}{\sin\left\lbrack {\frac{\pi\; d}{c}\left( {{u_{k}{\sin(\theta)}} - {u_{k^{\prime}}{\sin\left( \theta_{r} \right)}}} \right)} \right\rbrack}}},} & (3.1)\end{matrix}$where the unit phasor q_(kk′) is defined by

$q_{{kk}^{\prime}} = {{\exp\left\lbrack {t\frac{\pi\; d}{c}\left( {{u_{k}{\sin(\theta)}} - {u_{k^{\prime}}{\sin\left( \theta_{r} \right)}}} \right)} \right\rbrack}.}$Thus,

${{F_{{kk}^{\prime}}(\theta)}}^{2} = {\frac{\sin\left\lbrack {M\frac{\pi\; d}{c}\left( {{u_{k}{\sin(\theta)}} - {u_{k^{\prime}}{\sin\left( \theta_{r} \right)}}} \right)} \right\rbrack}{\sin\left\lbrack {\frac{\pi\; d}{c}\left( {{u_{k}{\sin(\theta)}} - {u_{k^{\prime}}{\sin\left( \theta_{r} \right)}}} \right)} \right\rbrack}}^{2}$and

$\begin{matrix}{{U\left( {\overset{\rightarrow}{R},t} \right)} \propto {{{tria}\left( \frac{t}{T_{p}} \right)}{\sum\limits_{k}{\sum\limits_{k^{\prime}}\begin{Bmatrix}{{F_{{kk}^{\prime}}(\theta)}\sin\;{c\left\lbrack {\left( {u_{k} - u_{k^{\prime}}} \right)\left( {T_{p} - {{t}A_{k}{\overset{\_}{A}}_{k^{\prime}} \times}} \right.} \right.}} \\{{\mathbb{e}}^{{\mathbb{i}}\; 2\;\pi\;{u_{k}{({{\delta\;{\tau\;}_{k}} - {\delta\;\tau_{0k}}})}}}{\mathbb{e}}^{{\mathbb{i}}\; 2\pi\;{u_{k}{({{\delta\tau}_{k} - \delta_{k^{\prime}}})}}t}}\end{Bmatrix}}}}} & (3.2)\end{matrix}$

When all the frequencies are identical, u_(k)=u₀=c/λ₀, then the arrayfactor

${F_{{kk}^{\prime}}(\theta)} = {{F_{\infty}(\theta)} = {q_{\infty}^{- M}\frac{\sin\left\lbrack {M\frac{\pi\; d}{\lambda_{0}}\left( {{\sin(\theta)} - {\sin\left( \theta_{r} \right)}} \right)} \right\rbrack}{\sin\left\lbrack {\frac{\pi\; d}{\lambda_{0}}\left( {{\sin(\theta)} - {\sin\left( \theta_{r} \right)}} \right)} \right\rbrack}}}$is a common multiplier in all the summation terms with which thecross-correlation is also proportional, a standard result for theconventional radar.

Otherwise, the array factor F_(kk′)(θ) is not only a function of thebeam steering angle θ, but is also dependent on the frequency paircombinations u_(k), u_(k′). As can be seen from (3.2) every term istapered by the array factor. In other words, each transmission amplitudeA_(k) is shaped by the antenna pattern corresponding to that frequency.Thus, the beam is realized as is required during reception.

Methods of Operation

FIG. 7 is a flow chart illustrating an exemplary method 700 fortransmitting and receiving radar signals from a radar system. Method 700includes blocks 702, 704, and 706. The order in which the method isdescribed is not intended to be construed as a limitation. Furthermore,the method 700 can be implemented in any suitable hardware, software,firmware, or combination thereof. In the exemplary implementation,method 700 is executed by antenna elements in conjunction with theexemplary components described above.

In block 702 a pattern of radar signals having different frequencies aretransmitted, simultaneously. For example, a transmission unit 322(FIG. 1) transmits the pattern 108 of radar signals F₁, F₂, F₃, F₄, F₅,F₆, F₇, . . . , F_(N) of different frequencies.

In block 704, the pattern is received after being reflected from atarget. For example, a receiver system 106 (FIG. 1) receives areflection of the pattern (i.e., see FIG. 1). A common referenceoscillator signal (such as 301 shown in FIG. 3) is used to phase lockthe transmission and reception of radar signals.

In block 706, the plurality of radar signals F₁, F₂, F₃, F₄, F₅, F₆, F₇,. . . , F_(N) having different frequencies are combined into a compositewaveform to form an image of a target. For example, a processing system,such as one or more processors 304 and 316 (FIG. 3), combine the radarsignals into a composite waveform.

SUMMARY

In summary, described herein is a novel concept of electronicallyscanned phase array (ESPA) radar whose operation employs a set of fullycoherent oscillators that are individually phase locked to a commonunmodulated reference oscillator but otherwise the oscillators may andwill operate at different frequencies. In fact, each antenna arrayelement is to transmit only a pure tone whose frequency varies from oneradiating element to another and from pulse-to-pulse but receives thereflected composite aggregate of such tones. (This is a simplificationbut not necessary to the concept; there are other ways of decomposing asignal, e.g., orthogonal polynomials, etc.) The aggregate of thesetransmitted tones is to cover the full radar bandwidth the latter beingcommensurate with the specified range resolution.

In one implementation, a waveform may be designed to satisfy somerequired range resolution and dynamic range requirement. Then thiswaveform is decomposed into a convenient set of constituents, such asFourier components. The latter having the advantage of possessingconstant envelopes but others are possible, too. These constituents aretransmitted in separate time epochs at the several radiators such a waythat each radiator is to transmit the full set of constituentseventually.)

The reflected composite signal is phase-coherently demodulated, that iscorrelated, at each element and combined. This coherent correlation overthe time and antenna angle variables will simultaneously compress therange and also form a synthetic spatially selective antenna beam. Byproperly adjusting the amplitudes and phases of the transmitted tones itis possible to shape and scan the antenna beam, respectively. Using thesame amplitude and phase variables, each receiver element locallyreconstructs a waveform that is expected at that particular elementlocation and coherently adds the result of correlation to maximize thesignal to noise ratio and minimize the antenna sidelobes.

The radar system described herein is, therefore, “spread spectrum” inboth between time and frequency spectrum domains and between angle andwave vector spectrum domains. (The individual antenna elements arealmost omni-directional, i.e., wide spectrum in wave vector domain; inthis sense this is synthetic aperture technique. At any given time,though, the elements radiate different waveforms, but they all receivecoherently the same composite one.) By permuting the frequencies of theradiating elements from pulse-to-pulse, the off-boresight antennapattern will strongly fluctuate, the waveform will scintillate, tocombat jamming. Also, since beam forming happens at signal reception andnot during transmission, the radar system has an inherently lowprobability of intercept.

Although the invention has been described in language specific tostructural features and/or methodological acts, it is to be understoodthat the invention defined in the appended claims is not necessarilylimited to the specific features or acts described. Rather, the specificfeatures and acts are disclosed as exemplary forms of implementing theclaimed invention.

1. A radar system, comprising: a transmission system, comprising aplurality of radiators, configured to transmit, simultaneously, aplurality of radar signals having different frequencies at a target andwherein each radiator is configured to transmit a radar signal that isunique with respect to other radar signals transmitted by otherradiators of the plurality of radiators, and wherein the transmissionsystem is further configured to permute the radar signals transmitted byeach of the radiators over time; and a receiver system, comprising aplurality of radiators, configured to receive at least a portion of theplurality of radar signals being reflected from the target, and tocombine the at least a portion of the plurality of radar signals into acomposite waveform representing an image of the target.
 2. The radarsystem as recited in claim 1, wherein the transmission system and thereceiver system are phased-locked to a common reference oscillatorsignal.
 3. A radar system, comprising: a common reference oscillatorsignal; a plurality of transmit/receive (T/R) elements phased locked tothe common reference oscillator signal, each configured tosimultaneously transmit a radar signal with a frequency that is uniqueto the particular T/R element and permuted over time, and simultaneouslyreceive a conglomeration of the radar signals reflected from a target,including the radar signal with the frequency that is unique to theparticular T/R element and permuted over time; and a processing system,configured to correlate each of the unique radar signals reflected fromthe target to each of the T/R elements that transmitted the radarsignal, and based on the correlation, combine the radar signals to forman image of the target.
 4. The radar system as recited in claim 3,further comprising a reference oscillator configured to generate thecommon reference oscillator signal.
 5. The radar system as recited inclaim 3, wherein each of the T/R elements includes a transmission unitand a receiver unit.
 6. The radar system as recited in claim 3, whereineach of the T/R elements includes a transmission unit, a receiver unit,and a processing unit.
 7. The radar system as recited in claim 3,wherein the processing system includes a plurality of processing unitseach corresponding to the plurality of T/R elements.
 8. A method fortransmitting and receiving radar signals, comprising: transmitting apattern comprising a plurality of unique radar signals having different,time permuted frequencies from a plurality of radiators simultaneously;and receiving a reflection of the pattern; and combining the pluralityof unique radar signals having different, time permuted frequencies intoa composite waveform to form an image of a target.
 9. The method asrecited in claim 8, wherein a transmission system transmits the patternand a receiver system receives the reflection and combines the pluralityof radar signals.
 10. The method as recited in claim 8, furthercomprising generating a common reference oscillator signal.
 11. Themethod as recited in claim 8, further comprising generating a commonreference oscillator signal, and phase locking the plurality of radarsignals when transmitting and combining the radar signals.
 12. Themethod as recited in claim 8, further comprising transmitting a radarsignal that is unique with respect to other radar signals.
 13. A radarsystem, comprising: means, having a plurality of transmit elements, fortransmitting a pattern comprising a plurality of unique radar signalshaving different, time permuted frequencies simultaneously; and meanshaving a plurality of receive elements for receiving a reflection of thepattern, and combining the plurality of unique radar signals havingdifferent, time permuted frequencies into a composite waveform to forman image of a target.